Edward Frenkel on education and inflation

Mathematics professor Edward Frenkel wrote an opinion Don’t Let Economists and Politicians Hack Your Math. Of course kids need to learn algebra, Slate Feb. 8 2013. This appears partly a response to Andrew Hacker,Is Algebra Necessary?“, The New York Times, July 28, 2012. Frenkel also refers to Matthew Yglesias, “CPI Unchained. The sneaky plan to cut Social Security and raise taxes by changing how inflation is calculated“, Slate December 30 2012, where CPI = Consumer Price index.

Let me respond as both an econometrician and teacher of mathematics.

First some facts

  • In Holland, kids are allowed to graduate from middle school without algebra. The system allows for various competences. Holland isn’t perfect, though. Apparently, in the USA you either learn algebra or you drop out. If there are no alternatives then this is a rather sick approach, and then Hacker is right. See this Journal of Humanistic Mathematics that discusses the wider ranges of competences.
  • Algebra builds upon arithmetic. Thus the source of the problems in the USA may be elementary school. See for example here.
  • Hacker has a nice example:
    “But there’s no evidence that being able to prove (x² + y²)² = (x² – y²)² + (2xy)² leads to more credible political opinions or social analysis.”
    The example is quite nice (find the trick), and again Hacker is right that this kind of proficiency has limited value. We should have people who can do this. For others it might suffice that they know how to handle and understand a computer algebra package like Mathematica or open source Sage.
  • Hacker has another nice example:
    “It could, for example, teach students how the Consumer Price Index is computed, what is included and how each item in the index is weighted — and include discussion about which items should be included and what weights they should be given.”
    Below I will give an example of what would be useful to people to understand.
  • The issue is mathematics education and not merely mathematics. Hacker didn’t study mathematics education but neither did Frenkel. The opinion pieces are a bit like priests discussing marriage.
    People are entitled to some opinion as to how competent professionals in areas of medicine or computers should be (“deliver what you advertise”), but it is a different issue to translate such levels of competence to the educational programme to get there.
    It is quite possible that Hacker would be very happy with algebra in schools if the system of education would be improved such that having algebra didn’t cause those dropouts, and generated the overall competence that he desires. Thus we should rather not discuss algebra but the quality of mathematics education – which quality is awful.
  • Frenkel, without his study of mathematics education, clearly misses the latter point, and starts defending the beauty and relevance of algebra. He is a typical mathematics professor who contributes to the dismal state of mathematics education by failing to understand that education is an empirical issue and not a playground for abstraction. Frenkel also abuses the CPI issue to scare the wits out of you, see below.

Thus, the sensible position has already been formulated in my book “Elegance with Substance(2009). Blame the mathematicians and the Slate and The New York Times journalists for not paying attention.

PM. Miles Kimball & Noah Smith remind us that the ability to do math is widespread, and that it is a myth that genes cause that there are people who can do math and people who cannot do math. They rightly point out that this myth causes needless suffering and must be gotten rid of. They refer to Hacker’s article as carrying the risk of furthering that myth. I don’t think that Hacker’s article and proposal carry that risk. While we should  work hard to improve math education and eliminate that myth, it remains that at some point in life a test will determine whether a pupil has mastered the subject or not. Rather than forcing the issue, it can be better for the pupil to let him or her proceed with the other competences and joys. Educational paths shouldn’t fix life thereafter but allow for flexibility later on too.

A quantity (volume) and price index

The following method has been in use at the Dutch Central Planning Bureau (CPB) for perhaps 60 years and provides a sensible and consistent way to navigate in complex waters. It uses the Laspeyres volume and the Paasche price indices.

  • Advantages of this are:
    • Volume and price changes add up to value changes.
    • The weights for volumes and prices are adjusted over time.
    • The choice of a different base period (for chaining) does not lead to different values for the aggregate data.
  • A (“disadvantage”) consequence however is that the aggregate chained volume index differs from the sum of the disaggregate chained volume indices. This however cannot be seen as a “disadvantage” since there is no reason why that sum should hold.

Consider a fictitious example of indices for the national old age pension (social security) for the age 65+. Suppose that the pension allows for (1) bread, (2) coffee and (3) some health insurance. This is called the “basket”. Then we calculate: (4) the aggregate volume and price indices. The outlays change from year to year, partly because of entrants and departures, and partly because of changing tastes and reactions to price changes.

We will use these symbols for a fictitious change from 2013 to 2014:

Price, volume and value, symbols for level and absolute and relative changes

Price, volume and value, symbols for level and absolute and relative changes

Let us assume that an average pensioner eats 100 loaves of bread per annum, at EUR 1 per loaf (wholesale price) and that this quantity changes by 5%. Let the price remain constant. This gives us the first line of the table below. Expenditure (wn) changes from EUR 100 to 105.

In the second line we record that 450 cups of coffee were consumed in 2013 at EUR 1.3 per cup, thus at a total expenditure (wn) of EUR 585. Due to a VAT change affecting coffee only, the price rises by 7.7% to EUR 1.5 and hence we see a drop in consumption of 4.4% to 430 cups of coffee per year. Expenditure on this item changes from EUR 585 to EUR 602.

In the third line we assume that the pensioner has medical attention for one hour per week, thus 52 hours per annum, at the cost of EUR 200 per hour. Thus the expenditure in 2013 is EUR 10400. In 2014 due to aging, the average pensioner requires 53 hours per year, and the medical cost rises to EUR 210 per hour (5%). The outlays from medical insurance rises to EUR 11130 per annum.

Combining these data, the total expenditure per pensioner is EUR 11085 in 2013 and rises to EUR 11837 in 2014. The “aggregate volume index” (2013 = 100) is 101.6%. The “aggregate price index” is 105.1%. The “aggregate value index” can be found by properly multiplying these, and is consistent with calculating 11837 / 11085 * 100.

Line 4 gives the aggregate value, volume and price of lines 1 - 3

Line 4 gives the aggregate of lines 1 – 3, in value, volume and price

(Calculated with c = ChainIndexTable[{{100., 105.}, {450., 430.} , {52., 53.}}, {{1., 1.}, {1.30, 1.40}, {200., 210.}}] in The Economics Pack.)

The method uses only arithmetic but the highschool student has to understand quite a lot of concepts: value, volume, price, level, absolute and percentage change, averaging, using different weights. The student can show insight in that the figures make some sense (a price rise causes a quantity drop), and that the aggregate outcome is dominated by the large component. The student will also see that when more of such calculations are combined (say pensioners, families with and without children, singles; with subaggregates in the provinces or states) then the system remains consistent, so that the method of aggregation does not introduce curious distortions.

The aggregate volume and price changes for baskets of single years can be “chained” by multiplying the annual results. The aggregate price index in 2014 with base (2010 = 100) would be: 100 * (1 + pp[2011] / 100) * … * (1 + pp[2014] / 100). The advantage of this chaining is that the basket is not fixed but follows actual expenditure.

A criticism is: since above VAT rise reduces real spending power, then there is more weight upon less real consumption. The above allows pensioners a rise of expenditure from EUR 11085 to EUR 11837, but “adequate compensation” should require EUR 26 more per person because of their induced loss of their cups of coffee. However, is this statistics or politics ? In terms of policy, giving EUR 26 more does not guarantee that this is spent on coffee. In terms of statistics, there is a choice of using weights of the baskets of either 2013 or 2014. The method of using Layspeyers volumes and Paasche prices allows to calculate the result for 2014 using the weights from 2014.

The role of statisticians is to also inform policy makers about the observed changes in the basket. The choice of the basket is politics.

Thus, one would agree with Hacker that it would be useful that this method would be taught in highschool in Holland, so that students know what their CPB is doing. Similarly for the USA. Yglesias refers to Jill Leyland (2011) for the UK.

In Holland, Bert Balk has pointed to the properties of the Divisia – Törnqvist index but hasn’t convinced others, or me, yet that its modestly larger computational complexity generates better results (and other criteria to select than the above). It is getting to be used more often in productivity measurement, but note that hours worked and wage inflation can also be aggregated in above manner by distinguishing levels of productivity and associated wages (and it is curious that this often isn’t done).

Frenkel’s embrace of conspiracy theory

The issue is also discussed in this Wikipedia article that at the time of writing this is not yet sabotaged by MIT students and their formulas. Yglesias referred to above observes:

“(..) the point is that there’s no unique right or wrong answer (…) for how to treat product shifting, and its impact on individuals’ welfare will vary enormously. This matters a great deal for Social Security, however, because benefit levels are adjusted upward each year in line with inflation. If Congress decides that chained index is the “right” measure of inflation, benefit levels will be lower than currently predicted and the deficit will go down.”

The answer is of course that social security should not be indexed on only inflation but on the general rise in welfare. (In the above: on wn. This should also hold for tax brackets.)  If there is a rise in productivity, then let also the elderly benefit from it. If everyone has a computer then let also senior citizens be allowed to get one. Thus, we have an easy issue of policy making here.

This is also what Yglesias concludes:

“The central point is that it’s politics all the way down. (…) glib talk about apples and “better” inflation calculations masks a policy that if done sloppily could be quite damaging to the low-income elderly.”

However, mathematics professor Edward Frenkel claims that you can only understand the issue when you have learned algebra, while preferably you should also know gauge theory. In his opinion (his is an opinon piece) there is also a political conspiracy, supported by economists, to exploit your lack of understanding of mathematics, and keeping you there.

(1) Frenkel’s major claim:

“Is economics being used as science or as after-the-fact justification, much like economic statistics were manipulated in the Soviet Union? More importantly, is anyone paying attention? Are we willing to give government agents a free hand to keep changing this all-important formula [for CPI / TC] whenever it suits their political needs, simply because they think we won’t get the math?

(…) What seems to be completely lost on Hacker and authors of similar proposals is that the calculation of the CPI, as well as other evidence-based statistics, is in fact a difficult mathematical problem, which requires deep knowledge of all major branches of mathematics including … advanced algebra. Whether we like it or not, calculating CPI necessarily involves some abstract, arcane body of math. [Frenkel doesn’t seem to be aware of above practical solution. / TC]

(…) The inflation index must account for this, so we have to find a way to compare the baskets today and a year ago. This turns out to be a hard mathematical problem that has perplexed economists for more than a century and still hasn’t been completely solved. But even to begin talking about this problem, we need a language that would enable us to operate with symbolic quantities representing baskets and prices—and that’s the language of algebra! [Frenkel confuses above simple arithmetic and algebra with the somewhat more involved algebra that Hacker was speaking about. / TC]

(..) As Weatherall explains in his book, to implement a true cost-of-living index, one actually has to use the so-called “gauge theory.” This mathematics is at the foundation of a unified physical theory of three forces of nature: electromagnetism, the strong nuclear force, and the weak nuclear force. (Many Nobel Prizes have been awarded for the development of this unified theory; it was also used to predict the Higgs boson, the elusive elementary particle recently discovered at the Large Hadron Collider under Geneva.) 

(…) So that’s where we find ourselves today: Politicians are still eager to exploit backdoor mathematical formulas for their political needs, economists are still willing to play along, and no one seems to care about finding a scientifically sound solution to the inflation index problem using adequate mathematics. And the public—well, very few people are paying attention. And if we follow Hacker’s prescriptions and further dumb down our math education, there won’t be anyone left to understand what’s happening behind closed doors.” [Well, there is a huge economic literature on indexation. / TC]

(2) Frenkel alarms us – a call to arms – shifting the blame to politicians and economists instead seeing his own fault:

“Now is the time not to reduce math curriculum at schools, but to expand it, taking advantage of new tools in education: computers, iPads, the wider dissemination of knowledge through the Internet. Kids become computer literate much earlier these days, and they can now learn mathematical concepts faster and more efficiently than any previous generation. But they have to be pointed in the right direction by teachers who inspire them to think big. This can only be achieved if math is not treated as a chore and teachers are not forced to spend countless hours in preparation for standardized tests. Math professionals also have a role to play: Schools should invite them to help teachers unlock the infinite possibilities of mathematics to students, to show how a mathematical formula can be useful in the real world and also be elegant and beautiful, like a painting, a poem, or a piece of music.”

It is hard to see a political conspiracy to block the education of mathematics. Kids and students already spent an amazing number of hours on arithmetic and other mathematics. It are the mathematicians and the abstractly trained teachers of mathematics who are the spoil-sports. They do not see that education is an empirical issue, and they insert their hobbies and traditions into their allotted time, without wondering whether they shouldn’t pay attention to their students. As said, Frenkel hasn’t studied education, so he is in breach of the integrity of science, claiming expertise that he doesn’t have.

(3) Frenkel claims a conflict of competence / interest for Hacker:

“In his book, Weatherall made an admirable effort to start a serious conversation about the need for a new mathematical theory of the CPI. But guess who reviewed this book in the New York Review of Books? Andrew “we don’t need no algebra” Hacker! There is nothing wrong with healthy debate; it can only be encouraged. But something is wrong when an opinionated individual who has demonstrated total ignorance of a subject matter gets called on over and over again as an expert on that subject.”

I haven’t read that review yet, but it seems to me that Frenkel wrongly disqualifies Hacker. The label “Hack” in Frenkel’s opinion piece would be like someone to write about “A mathematical proof of conspiracy by Dr. Frenkelstein”.

(4) Frenkel claims that there was already unscientific manipulation in the Boskin commission 1996:

But what most people don’t realize is that something similar had already happened in the past. A new book, The Physics of Wall Street by James Weatherall, tells that story: In 1996, five economists, known as the Boskin Commission, were tasked with saving the government $1 trillion. They observed that if the CPI were lowered by 1.1 percent, then a $1 trillion could indeed be saved over the coming decade. So what did they do? They proposed a way to alter the formula that would lower the CPI by exactly that amount!

(…) The fact that gauge theory also underlies economics was a groundbreaking discovery made by the economist Pia Malaney and mathematical physicist Eric Weinstein around the time of the Boskin Commission. Malaney, who was at the time an economics Ph.D. student at Harvard, tried to convey the importance of this theory for the index problem to the Harvard professor Dale Jorgenson, one of the members of the Boskin Commission, but to no avail. In fact, Jorgenson responded by throwing her out of his office. Only recently, George Soros’ Institute for New Economic Thinking finally gave Malaney and Weinstein long overdue recognition and is supporting their research. But their work still remains largely ignored by economists.

Mark Thoma had some comments on this in 2006 and quotes BLS in 2008. I am not tempted to follow this up here. I suppose that there is no conflict of interest in Frenkel and Weinstein appearing jointly at the Speyer Legacy school, both apparently without a background in the education in mathematics.

Overall, I am happy to include this subject in a parliamentary enquiry on unemployment and/or another such enquiry on the dismal state of mathematics education.

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