Our World: Market Rebounds after Assurances from Fed Chair
Mathematically Literate World: Market Rebounds without Clear Causal Explanation
Our World: Firm’s Meteoric Rise Explained by Daring Strategy, Bold Leadership
Mathematically Literate World: Firm’s Meteoric Rise Explained by Good Luck, Selection Bias
Our World: Gas Prices Hit Record High (Unadjusted for Inflation)
Mathematically Literate World: Gas Prices Hit Record High (In a Vacuous, Meaningless Sense)
Our World: Psychologists Tout Surprising New Findings
Mathematically Literate World: Psychologists Promise to Replicate Surprising New Findings Before Touting Them
Our World: After Switch in Standardized Tests, Scores Drop
Mathematically Literate World: After Switch in Standardized Tests, Scores No Longer Directly Comparable
Our World: Controversial Program Would Cost $50 Million in Taxpayer Money
Mathematically Literate World: Controversial Program Would Cost 0.001% of Taxpayer Money
Our World: Proposal Would Tax $250,000-Earners at 40%
Mathematically Literate World: Proposal Would Tax $250,000-Earners’ Very Last Dollar, and That Dollar Alone, at 40%
Our World: Poll Finds 2016 Candidates Neck and Neck
Mathematically Literate World: Poll Finds 2016 Predictions Futile, Absurd
Our World: One Dead in Shark Attack; See Tips for Shark Safety Inside
Mathematically Literate World: One Dead in Tragic, Highly Unlikely Event; See Tips for Something Useful Inside
Our World: Local Heat Wave Seen as Sign of Global Warming
Mathematically Literate World: Local Heat Wave Not Seen as Meaningful Indicator of Global Trends
Our World: Veteran Baseball Player Enjoys Breakout Month
Mathematically Literate World: Veteran Baseball Player Enjoys Transient Good Fortune
Our World: Market Share for Electric Cars Triples
Mathematically Literate World: Market Share for Electric Cars Rises to 0.4%
Our World: Still No Broad Agreement on Global Warming
Mathematically Literate World: Still 90% Agreement on Global Warming
Our World: Rates of Cancer Approach Historic High
Mathematically Literate World: Rates of Surviving Long Enough to Develop Cancer Approach Historic High
Our World: Hollywood Breaks Box Office Records with Explosions, Rising Stars
Mathematically Literate World: Hollywood Breaks Box Office Records with Inflation, Rising Population
Our World: Economist: “Eliminate Minimum Wage to Create Jobs, Improve Economy”
Mathematically Literate World: Economist: “Eliminate Minimum Wage, then Pray Our Model Has Some Empirical Basis in Reality”
Our World: Average Football Player Earns $3 Million, Lasts 4 Years in NFL
Mathematically Literate World: Average Football Player Earns $0 Million, Lasts 4 Years in High School
Our World: Politician Promises to Fund Math Education
Mathematically Literate World: Politician Promises to Fund, Meddle in Math Education
Our World: Illegal Downloaders Would Have Spent $300 Million to Obtain Same Music Legally
Mathematically Literate World: Illegal Downloaders Probably Would Not Have Bothered to Obtain Same Music Legally
Our World: Unemployment Rate Jumps from 7.6% to 7.8%
Mathematically Literate World: Unemployment Rate Probably a Little Under 8%; Maybe Rising, or Not, Can’t Really Tell
Thanks for reading! (And thanks to readers who suggested improvements, some of which I’ve incorporated.) If you can tolerate my drawings (a big “if”), you might also enjoy A Math Professor Consults on a Hollywood Movie and the economic satire 20 Steps for Trading Up from a Paper Clip to a House.
You could also check out Fifty-Five Million, my new blog that aims to bring a math-literate perspective to education statistics. And I highly recommend Aaron Brown’s excellent critique of this post, delving more deeply into some key issues.
Math with Bad Drawings 
How is 0.0001% more literate than $50 million? That just seems like an arbitrary choice
Fair point. The literate person would know both figures, not prioritize one over the other.
The idea was mostly that people tend to be dazzled by numbers ending in -illion, when really, some -illions are much bigger than others.
Your initial instinct to include that example was spot-on. Part of literate writing is putting numbers in a context that readers will understand. Assuming that everyone is familiar with the specifics of the federal budget (or the state/local budget) is bad writing. Glossing over those specifics to deliver sensationalist headlines is worse writing.
This is why I advocate all discussion of federal spending to be in units of a trillion dollars. (States could use billions.)
I like the percentage metric (or denominating them in trillions), but I think your number is off. It should be 0.001%, not 0.0001%.
I think quite a useful thing to know is the cost divided by the population (or perhaps by the number of households, or the number of taxpayers). That way, one can think, “Is this worth what I’m spending on it?” Of course, the point then has to be made that even if it’s not worth it to you, other things are worth it to you that are not worth it to other taxpayers who are contributing to them.
But the initial choice of words isn’t arbitrary either. If the government has it, it’s no longer “taxpayer money”, and it doesn’t necessarily come from tax income (there are other forms of government income, http://en.wikipedia.org/wiki/Government_revenues).
With that in mind, it seems clear to me that the combination of the words “taxpayer money” and a seemingly large figure is a hint that the program may be a waste of money. Putting it in perspective with the entire government budget seems like an attempt at fairness with the facts.
Agreed – this isn’t a “mathematical” literacy question per se, but “federal budget” is a more honest term than “taxpayer money.” (After all, I’m a taxpayer, but I don’t pay all my money in taxes!)
Also, as with several of these, sometimes the second one in the pair serves less to replace the original than to comment and elaborate on it.
I’m curious about “Proposal Would Tax $250,000-Earners’ Very Last Dollar, and That Dollar Alone, at 40%” — not living in a world reported on by mathematically literate media, that one was news to me. Help?
Hmmm… to start, picture a simplified two-tax-bracket world, where income below $250k is taxed at 10%, and income above $250k is taxed at 40%.
If you earn $250k, all your income is taxed at 10%, so you pay $25k.
But if you earn $250,001, you don’t pay 40% on ALL the income. You only pay 40% on that last dollar earned. So your total tax is $25,000.40.
In our real tax system, the rate for income below $250k is much higher than 10%. But the principle is the same. Raising taxes on income above $X won’t affect you very much at all if you make only a little more than $X.
Indeed, this is the basis of the common lie “Our tax system is so broken, if I made more money, I’d take home less.” The only way for that to happen is for the marginal tax rate to rise over 100%.
I think it’s more the welfare system. When they cut your unemployment benefits so much that a really low paying job might mean you take home less.
This was not always the case. Before the 80’s tax reform, the brackets and rates were so complicated, you *could* take home less with a higher gross (my dad had this happen to a secretary). Not true anymore, though there could be some narrow windows with 100+% marginal rates with recent tax law changes.
Wonderful post, BTW.
Nate Silver did a post about a year ago on a proposal that would have created marginal tax rates above 100%. Thankfully the idea didn’t go anywhere.
http://fivethirtyeight.blogs.nytimes.com/2012/11/26/congressional-proposal-could-create-bubble-in-tax-code/
Well, this applies if there are people earning exactly $250,001. The correct statement is that any _earned_ income above $250k will be taxed at 40% by the federal government.
Did you even bother reading the statement being corrected?!
The idea in play here is “marginal tax rate”. Proposals concerning taxes for the wealthy are usually about taxes on their *marginal* income (above some threshhold), but are often reported as if they are a tax rate on their entire income, full stop.
To use Ben’s example — imagine if you *did* get kicked into a 40% tax rate on your whole income, instead of your marginal income, once you made 250k! You would go from making 249,999, paying 24,999 in taxes, and bringing home 225,000…to making 250,000, paying 100,000 in taxes, and bringing home 150,000. And that would be *totally stupid* (not to mention an inducement to tax fraud).
So we don’t do things that way. But reportage tends not to draw the marginal-tax-rate distinction.
This is why I can’t read the newspaper, or especially watch TV, without getting mad.
“How to Lie With Statistics” is a book that never goes out of style.
Sadly true. I think most media tilts towards sensationalism more than any particular partisan bias, but that’s not a great situation either.
One *minor* wrinkle 🙂 At least worldwide, (some of) the rise in cancer rates is *probably* not an artifact of people simply living long enough to get cancer; that some countries in the so-called developing / underdeveloped world, with very low life-expectancies, are seeing what looks to epidemiologists like a real rise in cancer rates (despite in some cases *falling* life expectancies!), points towards different causal pathways (increased tobacco use, the introduction of other environmental carcinogens, & changes in diet would seem to be the ‘big three’).
The broader lesson, nicely brought home by these examples, is of course that such headlines are generally misleading, and that digging into the epidemiological data is required to make any sense of them. But at least in cancer research, my reading of the epidemiological literature is that there are not *many* cancers where a) rates are generally agreed to really be rising (not an artifact of better screening, etc.) and b) the rise is mostly attributable mostly to people living longer.
A more subtle question, harking back to some of the issues brought out in the Bear in the Moonlight. Say I flip two sets of 10 coins repeatedly. One set is fairly strongly biased towards heads, the other set is fair. Say I’m considering a particular flip of a set in which I’ve gotten 8 heads. Does my explanation of why I got 8 heads, in this particular flip, depend on whether I’ve flipped the biased or the unbiased set? In both cases, should I just note that each individual coin’s particular outcome was determined by e.g. its physical features plus the particular factors that went into its toss, and hence that any particular outcome can only be explained on the basis of those kinds of individual factors, or am I permitted, in explaining a particular outcome, to appeal to features of the kinds of the distributions that are expected? (Does this change if some outcomes are impossible or guaranteed, rather than merely more or less likely, given the different sets of coins?) Does the answer one gives here bear on (!) the question of whether global climate change could ever be legitimately invoked to “explain” the occurrence of any individual-level weather event (where that particular kind of event could have occurred in the absence of the global change, even if it was much less likely to)? (I wish to take no stand here on whether there have been actual individual weather events that fit this pattern. The question is rather about your views on the legitimacy of appealing to distributions to explain individual outcomes more generally.)
I would say that distribution-level explanations never apply to individual events.
Those are some great questions!
On cancer rates, my knowledge of the epidemiological literature is zero, so it’s interesting to learn that rising cancer rates (where they exist) aren’t attributable to demographic changes.
As for distribution-level explanations for individual events, I think there are two separate issues here.
First, can a single data point (or a few data points) be seen as an indicator of a global trend? My philosophy is, “Maybe yes, but proceed with caution.” It seems fair to take storms like Sandy and Irene as tentative evidence of larger shifts in weather patterns. But I wouldn’t accept two ordinary rainstorms as remotely compelling evidence of climate change. In the case of hurricanes, a sample size of two is actually pretty big. In the case of rainstorms, it isn’t. In short: I won’t categorically rule out drawing inferences off of small sample sizes, but context matters.
Second, can a global trend be used to explain a single observation? This seems to boil down to a personal question of what explanations we each find satisfying. For myself, I often do find distribution-level explanations satisfactory. “Whoa, how did he make two three-pointers in a row?” “Well, he shoots 45% from behind the line.” Sure, in theory we could appeal to the tiny differences in physical circumstances governing the trajectory of the basketball… but in practice, we virtually never have that level of information, so it seems silly to require it for explanations.
Hi Ben —
The problem with moving from explanations at the level of distributions to individual explanations (and I suspect why John Cowan suggests that such explanations are never legitimate) is that “Well, he shoots 45% from behind the line” could be an explanation for why he made two 3 pointers in a row, why he *missed* two 3 pointers in a row, or why he made one and missed one.
In each case, the “right” answer (statistically) seems to be “well, given the odds of making a 3 point shot, this actual observed distribution is not unexpected.” If we moved to missing, say, 6 shots in a row, we would have to say “this distribution, while unexpected, is not impossible, and given enough time, we were sure to observe something like it; it can still be explained by the ordinary odds, etc.”
Certainly, knowledge of the distributions can tell us what is *possible.* And why something is possible is surely part of the explanation of why it occurred. So that he shoots 45% from behind the line is a good explanation for making two shots in a row, insofar as the contrast class is say, me — I shoot pretty near to 0% behind the line! But this of course points towards our explanations being explanations for a purpose, within a particular set of contrast classes, etc. And that begins to point towards some of the ways in which our explanations rely on context…
So we might naturally think that moving from individual events to trends is “just” a matter of straightforward Bayesian inference 🙂 We just have to know how likely the observed event would be given different possible states of world, etc., and do the math.
But this means that a storm that we cannot, in good conscience, attribute to global warming (say), might yet provide evidence of global warming, if we are more likely to observe storms like that given GW than without it. Similarly, we might not be able to explain any particular outcome of a 10-toss coin flip on the basis of the coins being biased (all we would be able to say is that given some particular bias, we are likely to see such and such outcomes every so often, etc.), but a particular outcome could yet provide evidence of bias.
Moving back and forth between these, however, we should get to a point where I would think we would want to explain particular outcomes as a consequence of the actual state of the world. Why did we get this particular outcome? Any number of possible states of the world are compatible with this particular outcome, but given past actual outcomes, we have good reason to think the actual state of the world is this, and hence that the actual state of the world ‘explains’ the particular outcome we got. Or have I cheated and shifted the sense of ‘explanation’ in there? 🙂
*applause*
*awkward bow*
Our World: Authoritative Voice: “X% of scientists agree with Y”
Mathematically Literate World: With ‘Y’ loosely defined, ‘agree’ being a binary representation of a gradient, and the low value science places on consensus in the first place, this must be a Tide commercial.
Mmm, well said.
You think science places a low value on consensus? I’m not a practicing scientist, but it seems like science likes to attack the places where consensus is lacking.
I don’t know if this is what the OP meant, but, as a practicing scientist for the last ten years (yikes!), I can say that neither I nor any scientist I’ve talked to or observed has ever worried about the consensus view. We worry about good arguments, not about consensus.
There are, of course, arguments that are widely thought to be good and proposing an explanation which either does, or appears to, violate one of those will result in being pointedly questioned about that inconsistency. And this fact tends to result in very few proposed explanations that do, in fact, violate those arguments.
Given the dictionary definition of consensus, this could be called that, but it seems to suggest a more forceful sense of agreement and judgement than is actually the case. We do not get together and vote on The Answers to various questions; I do not even know what those *are*. We only have common sets of explanations for particular parts of the world, i.e. theories.
If by ‘lacking consensus’ you mean ‘no clearly good argument yet available’, then science does get interested in many such things. But it also specifically *avoids* many questions where there is no good argument. We ignore psychic powers and UFOs, for instance. At the same time, we tend to cluster around questions where a great deal *is* known. These days, for example, graphene comes to mind. Previously, it was carbon nanotubes (CNTs). At one point there were 400 papers CNTs coming out every month. And there were never enough controversial points about CNTs to account for 1% of that.
Most research answers questions that aren’t so much debated as they are simply as-yet-unasked. More than once I’ve had a new grad student come to me after first starting work on a question and ask what book they should read to know what people think about the issue at hand, to which I answer:
There isn’t any book to read; no one knows; that’s why you’re looking into it.
Questions of consensus are clearly irrelevant to this situation, which is my main point. We don’t place either a high or a low or even a moderate value on consensus, we simply don’t concern ourselves with that aspect of the situation.
Thanks – that’s a nice exposition.
The point of the original joke was that conservative media often use the presence of scientific disagreement about anthropogenic explanations for climate change as a sign that there is NO agreement, when in fact, the presence of some disagreement in a scientific community doesn’t mean there isn’t a broadly accepted explanatory framework.
Since media do this with all kinds of things, not just science, maybe a better alternative would have been something like:
“Still no agreement on hot-button political issue”
vs.
“Still 90% agreement on media-fabricated political issue.”
wow, yes, exactly that : ). While I’m sure consensus is taken into account when thinking about a problem, it has no place in an argument or a proof (the science part). Also the scientific consensus usually can pivot fairly quickly with new evidence or experiments – eg ‘are continents drifting’, ‘are Mayan glyphs writing’ etc.
Mostly when I hear that kind of thing I assume it is for ‘marketing’ purposes and the terms are generously defined to make any point desired. But I suppose sometimes it is just out of laziness (or insufficient expertise) to explain. I’d way rather hear about the reasoning and results though.
Love your list btw, many an argh moment in there : ).
It’s been interesting hearing science’s take on those “4 out of 5 scientists say…” headlines. I guess this is why we’ve got science writers–so they can do some justice to the arguments for a lay audience, rather than just take an opinion poll of scientists, which doesn’t really convey the nature of the issues.
Our World: Sensational Headline: “Scientists are baffled…”
Mathematically Literate World: Scientists are not baffled.
Average football player makes $0, lasts 4 years in high school? False. If ANYONE ever makes money in football, then the average money that football players make is nonzero. Perhaps you meant to say median?
perhaps you both meant to say modal?
Well, if you *want* to be pedantic about it…
The number he quoted was $0 million, with one significant figure. Which means it’s an accurate representation of any amount below five-hundred thousand dollars.
A couple quick google searches indicates the mean earnings of football players (HS thru NFL) to be around $4,000 per year.
That’s based on 2K NFL players making a mean of $2M per year and around 1M HS players.
I like this discussion!
Anyway, usually when we say “average” we’re referring to the mean, but I’ve also heard “average” used for other measures of central tendency (specifically median and mode). In a case like this, where the mean is heavily influenced by dramatic outliers, it makes sense to go for one of the other two.
I’d interpret this not as the average earnings but the earnings of the average player, in which case $0 is exactly correct.
“average player” has no meaning in reality
Typically, when people say “the average person” they mean the median person.
How can a firm’s meteoric rise be attributed to good luck? We all know there is no such thing as luck, good or bad. Selection bias I agree with, but a firm could do better than others in its market sector by employing better strategies and having the skill to implement them effectively.
That’s fair. I guess it depends on perspective. Let’s say there are 10 firms trying 10 different strategies, and one of those strategies turns out to be wildly successful, but for reasons the firm didn’t anticipate. You can call that good strategizing, but I’m also happy to call it luck.
There’s also the luck based elements of the individual people that make up the firm knowing or meeting the right people, which in turn can be based on getting the right first job, going to the right college, taking part in the right extra-curriculars in high school, having the right interests at a young age, being born to the right family, in the right country, being white, etc… all things that can have a lot to do with luck.
This is the purpose of the word serendipity. You don’t get good luck or bad luck in the sense that you are not a lucky or unlucky person, but you can experience a series of fortunate events or happy coincidences which could be viewed as good luck. Fortunately we have a single word to mean just that.
Great post!
Thanks!
Our World: Cancer sufferers living longer!
Mathematically Literate World: More non-aggressive cancers being discovered due to screening.
Oof. That’s a really interesting one–and a case where knowing the actual statistical story is a bit of a downer.
If shark attacks are highly unlikely, then shark safety tips are more useless than useful, no?
Yeah, true; I meant the “something useful” in contrast to “shark attack avoidance tips”–the idea being that in the mathematically literate world, they’d give advice for something more applicable, like car safety or cancer screenings.
Beautiful! Thank you so much!
You know, you should do another series on “one-liners from movies/TV as re-stated in a mathematically literate world”. E.g., the one which always leaves me *hopping mad*: the medical examiner looks up from murder victim’s corpse, says “liver temp 80 degrees, time of death: last night at 2:37am”; should read: “liver temp 80 degrees, now if I could solve one equation — and it would be only one equation under certain simplifying assumptions, like that the ambient temperature was constant for the whole night — for two unknowns (time since death, rate of heat transfer to the environment), in my head, without a calculator to do the logarithms, I would say that the time of death was sometime last night, one standard deviation being the hours between 1:30 and 3am.”
Lol. That’s great. I’d love to see them call in a specialist in partial differential equations to produce an answer like that.
As far as incentives to change behavior go, the tax rate on the last dollar is the important figure.
BTW, shouldn’t you also include something on absurdly-small sample sizes?
Yeah, I guess the only sample-size related one is the “local heat wave not seen as meaningful indicator of global trends” one.
Agreed that tax rate on last dollar is an important figure. But I’m always shocked at how few people understand the actual mechanics of tax brackets. “Marginal tax rate” is not a term most Americans seem to know or get the significance of, and I think the most common misconception is that the marginal tax rate applies retroactively to all money earned.
I have a small problem with the 90% consensus on global warming. First of all 90% consensus is meaningless in science. Only in Islam a dogma is “I will never lead my people’s majority into falsehood”. It is quite often the lonely not consenting scientist who turns out to be right (though quite often is also meaningless statistics, I admit). Second of all, there has never been and never will be consensus in mathematics, that is why we have axiomata. Even on the question of whether parallel lines will ever meet there has not been consensus, moreover, it has given us a beautiful new geometry. Lastly, all this “consensus” is based on computer models, which are not only prone to programming errors, but also depend completely on their input.
“Second of all, there has never been and never will be consensus in mathematics…”
Wow: mathematics is probably the greatest consensus-generating machine ever produced. There is 99.99% consensus achieved, repeatedly.
“Even on the question of whether parallel lines will ever meet there has not been consensus…”
No one “disagrees” about this! In different geometries the answer as to whether they do or don’t is different, but there is literally 100% consensus about what each of those geometries are like.
Yeah, Leen, I think you’ve got a fair critique of the global warming one–some folks above made similar comments.
As for mathematics, I’m with gcallah–once you’ve got a valid proof, there’s nothing to disagree about. Certainly there are exceptions, though: the ongoing debate over the axiom of choice, say, or the occasional blockbuster proof which is too monstrous and technical for anyone to actually check over.
A great read, a bit of obvious political bias in some of the topics selected. For example, the minimum wage one could have easily been about a hike to the minimum wage as well. That said I enjoyed it, I find myself frustrated by the use is stupid math in the media on a daily basis.
Yeah, guilty as charged. Although in the minimum wage case, classical economics would predict that a hike in the minimum wage would increase unemployment, and so critiquing that prediction instead would still place me on the liberal side of things.
For me, the problem with the minimum wage example was that your correction didn’t bring mathematical principles into play, but simply reflected your own, subjective, critique of the economic paradigm (a problem borne out by your counterexample above). Saying something isn’t mathematically provable shouldn’t be understood to mean it’s not a valid proposition – even in a mathematically literate world. E.g., “I’m going to open the door for every person I encounter today, and I’m going to be happier for having done so.”
The mathematical principle is this: “Models are great, but they’re only useful insofar as they’re predictive or explanatory of empirical reality.”
The classical econ model says that if you institute a price floor for a good, then less of the good will be sold. Empirically, that seems to occur in many markets. But from what I understand, the evidence in the labor market is mixed. Thus, people who say, “Eliminating the minimum wage will necessarily increase employment” are acting as if an elegant mathematical model can trump reality.
I’m not saying it WON’T increase employment. I’m saying the empirical evidence is mixed, and you would need more solid empirical support to justify a claim like the one in the original headline.
With that one, I wasn’t going for a political point, so much as an “economic models are often over-extended” point.
ONE minor quibble: 90% is not a “consensus”
For semantic purposes, you could say “a near consensus” but really you mean to say “vast majority” or just plain “majority”
Number one: The very first one is rather bad. Yes, the media often gives silly explanations of stock price movements. But “regression to the mean” is not ever a causal explanation of anything. Is just a statistical tautology.
Aw, but I’m so proud of my portrait of Bernanke!
Anyway, I think there’s an underlying epistemological debate about whether stock price movements are always causally explainable. I’m open to the idea that in a random process, there might be no readily available and intellectually satisfying explanation of any particular event. So I’m often willing to settle for a post-hoc half-explanation like “regression to the mean.”
xkcd made a similar point about interpreting noise with respect to sports: http://xkcd.com/904/
Ben – by now you must be weary of compliments but I have to say this post is just brilliant.
Thanks! I love that xkcd.
Ben,
If you are going to take on the deceptive nature of reporting on tax rates, you should really get to the root of the problem, which unfortunately you perpetuate with your correction.
Your headline: Proposal Would Tax $250,000-Earners’ Very Last Dollar, and That Dollar Alone, at 40%
Accurate headline: Proposal Would Discriminate Against $250+k Earners By Applying Higher Effective Tax Rates On Them
The main problem with your headline is that in a mathematically literate world, people would know that individuals, not dollars, get taxed. The notion of taxing a marginal dollar earned is nothing but a deception devised to hide the discriminatory nature of a progressive income tax. No one actually thinks about his own tax burden in terms of marginal dollars earned. We think of our tax burden as an effective rate on our entire income for a year. Let’s take the simple example you used earlier, where income below $250k is taxed at 10%, and above $250k at 40%. A person who earns $200k will pay $20k in taxes, or 10% of his income. A person who earns $400k will pay $80k in taxes, or 20% of his income,
The higher earner is taxed at an entirely different rate than, and in fact double the rate of, the lower earner on ALL his income. The “marginal dollar earned” concept exists only to give the appearance of equality before the law – Everyone is taxed x% on their “first” dollar, and then y% on the “marginal” dollar above Z. But in a mathematically literate world, everyone would know that putting an income tax on a marginal dollar earned above a certain amount is just convenient way of legally discriminating against people who earn certain amounts of money, by charging them a different effective tax rate than everyone else.
Higher earners do face a higher effective tax rate. This effective tax rate will invariably be lower than their marginal tax rate (although as you earn more and more money, the effective rate will approach the marginal rate asymptotically). Many people do not seem to be aware that marginal and overall rates differ, which is why I chose to focus on the difference.
As for whether our system amounts to prejudicial treatment of those with higher incomes, that’s a political question, not a mathematical one. (And your claim that workers focus on overall tax, not marginal tax, is an empirical claim that I’m not sure is supported by the economics literature.)
Ben:
“This effective tax rate will invariably be lower than their marginal tax rate”
Yes…and it will also be invariably higher than the effective rate paid by everyone else who makes less money.
“Many people do not seem to be aware that marginal and overall rates differ, which is why I chose to focus on the difference.”
That is true enough, but you never actually mentioned “overall”, or effective, tax rates, which was why I commented.
“And your claim that workers focus on overall tax, not marginal tax, is an empirical claim that I’m not sure is supported by the economics literature.”
I don’t know of any economics literature that has addressed the subject of whether taxpayers view their tax burden in terms of an effective rate or a series of marginal rates, but I find it very difficult to believe that many people (or, really, anyone), if asked how much of their income they paid in taxes this year, would respond “Well, on my first $100k I paid X%, but then on my next $100k I paid Y%”. No one in my experience thinks of their taxes in that way at all.
Even the IRS tax tables don’t look at taxes in terms of marginal rates. For incomes below $100k they just provide a flat $ amount for a given amount of income, and for incomes above $100k, they give a single effective rate to multiply by the total income. No where can marginal rates be found. See here: http://www.irs.gov/pub/irs-pdf/i1040tt.pdf
Anyway, my original point was simply that, while you are correct to point out the error in thinking that a marginal rate applies to an individual’s total income, it is also an error to think that ” marginal dollars” get taxed. They don’t. Individuals get taxed, and they get taxed at a single, effective rate based on the amount of income they earn. The most accurate reporting on a change in tax law would not focus on how marginal rates are changing, but would instead focus on how effective rates are changing for certain, targeted demographics.
We seem to agree on the following facts:
(1) The more money you make, the higher your effective tax rate is.
(2) The effective tax rate is not the same as the marginal tax rate.
I agree that people should know both facts. But in my experience, most people know fact #1 already.
You argue that focusing on marginal tax rates is an error, or a dishonest representation of our system. I’m open to the idea that it’s a psychological error (i.e., it’s not indicative of how people actually think about taxes). But it isn’t a mathematical error. It’s an accurate mathematical description of how income tax works. There are other descriptions, too. But mathematically, they are equivalent.
I must disagree with #1 below. During the election I learned that Mitt Romeny’s effective tax rate was lower than my effective tax rate. Ben’s graphic was not at all deceptive.
That’s true about Romney – most of his income is classified as capital gains, which are taxed at 15% (rather than as salary or wages). But it’s something of a separate issue. I was aiming to focus on a progressive income tax system in general, rather than the tangled specifics of our tax code.
ScottC: “Even the IRS tax tables don’t look at taxes in terms of marginal rates. For incomes below $100k they just provide a flat $ amount for a given amount of income, and for incomes above $100k, they give a single effective rate to multiply by the total income. No where can marginal rates be found. See here: http://www.irs.gov/pub/irs-pdf/i1040tt.pdf ”
Where do you think the subtraction amount in column (d) comes from? That’s the correction factor for the different marginal rates set by statute. (Which, by the way, are on page 105 of that document, although they’re not referred to under that name.) This shortcut is a lot easier to apply than breaking up your income into pieces and applying different marginal rates to it, but it’s exactly equivalent in outcome. Let’s use Ben’s simple example with a base rate of 10% and a marginal rate of 40% after the first $250,000. On the worksheet for this tax system, you’d tell people making over $250,000 to calculate their tax by multiplying their net taxable income by the 40% marginal rate, and then subtracting $75,000; this is the amount the simple percentage overcharges for income below the margin, (40%-10%)*$250,000. The marginal tax rates aren’t mentioned in the instructions because doing it that way would make filing your tax returns even more horrendously complicated than it already is. But those worksheet instructions as well as the numbers in the tax tables are all calculated based on those marginal tax rates as set by Congress in the internal revenue code.
ScottC gives an example:
“Let’s take the simple example you used earlier, where income below $250k is taxed at 10%, and above $250k at 40%. A person who earns $200k will pay $20k in taxes, or 10% of his income. A person who earns $400k will pay $80k in taxes, or 20% of his income,”
I’m pretty sure that this is wrong and ignores the whole point of marginal tax rates. Our hypothetical earner pays 10% or $25k on the first 250k of her 400k income. On the next 150k , she pays 40% or 60k. So her total taxes on 400k are $85k, or 21.25% of her total income. In any case, it is nowhere near the 40% on TOTAL income that the scaremongering headline would suggest.
Of course, the real argument here is about the idea that marginal dollars eventually reach a point of decreasing utility and should be taxed accordingly. Or as some people would say, Enough is enough. At some level, money is no longer about the necessity of earning a living or even establishing a future for one’s family; it becomes a means of social display and dominance. Higher tax rates for higher income levels should not be punitive but should recognize the higher proportion of income devoted to necessity in lower-income households.
But for the real-life Gordon Geckos among us, there is no such thing as enough.
Reblogged this on Arijit Banik and commented:
When math and ‘art’ collide…
How about “Controversial Program Would Cost 0.157 cent to each American”?
btw, wolfram alpha is amazing:
http://www.wolframalpha.com/input/?i=50+million+%2F+USA+population
I agree with the original point that people do not have an intuitive grasp of the difference between a million, billion, or trillion dollars. (Whether this is something to lament or just accept as a basic fact of biology is an interesting question…) However, an increased understanding of this distinction using the cost per person, or cost per taxpayer, or percentage of the overall budget, etc., would be irrelevant to the political question. The implication is that people are foolish to object to a program that would only cost a tenth of a cent to each American. Just because a program uses a small fraction of the overall budget does not mean it is a good idea for the government to spend money on it. It may make it a lower priority to worry about than other programs, but there’s nothing intrinsically wrong with objecting to it, and the fact that it costs a small percentage is not an argument in favor of the program.
One could just as easily use examples from a conservative perspective, e.g. “national debt now $56,000 for every person in the country”, instead of 17 trillion. Or “Obamacare: it is not mathematically possible for 1) everyone to keep the plan they are on, 2) increase the number of people covered by insurance, 3) increase the amount of coverage required in insurance plans, and 4) have every family see a drop in premiums of $2500.”
I agree that cost per taxpayer, while interesting, is probably irrelevent to most issues. But I think percentage of budget is often a relevant factor. If I were voting in a referendum on a government program, I’d want to know, among other things: (1) What is the absolute cost? and (2) What share of the relevant budget will this consume?
Anyway, I think the takeaway is that one figure in isolation can mislead. Several numbers presented together can paint a more honest picture.
I like your drawings! I think that they really get the point across!
It is all how you look at it. Your examples are great. Which weighs more a ton of bricks or a ton of feathers?
Well, the ton of feathers will take up an awful lot more space!
It depends how dense the space is.
True. It’d be fun to have a super-compressed feather block that’s the size of a shoebox and the weight of a car. Although I don’t know where you’d keep it…
On the planet with the lightest density. It would cause a black hole and you could go back in time when feathers were on the dinosaurs..
I’m sold. Let’s go!
I am already there.
This was hysterical. Should be required reading in high schools, everywhere. I mark my mathematical savvy–and wry disdain for all commercials–from high school stat class. What an eye-opener THAT was. Hoo, boy. Four out of five dentists would agree.
Our World: Economist: “Eliminate Minimum Wage to Create Jobs, Improve Economy”
Mathematically Literate World: Economist: “Eliminate Minimum Wage, then Pray Our Model Has Some Basis in Reality”
I propose:
Mathematically literate world: The minimum wage is 0. It can’t be changed.
They could charge you to work for them, in which case there could be a negative wage, right?
See also: “internship for college credit.”
It’s an interesting point…. However I don’t think the IRS would look at it that way. Try entering a negative number on your 1040 form under “wages, tips and other compensation” and see what happens.
wow good drawing!! I love it.
Thanks–first time I’ve heard “good drawing”!
u do good blogs well done
I more or less agree with all of the points, but the one about minimum wage. I’m not saying whether eliminating the minimum wage will increase employment or improve the economy, but the ‘mathematical world’ response is very non-mathematical, it was much more politically biased.
The language (“pray,” “some basis in reality”) is a little loaded, but the idea that models need empirical verification is pretty apolitical.
True, but the economist may have had a model. Now you can disagree with the model, etc, but the “mathematical world” response doesn’t have any sort empirical retort in it at all.
To me, “basis in reality” means “empirical validity.” But obviously that didn’t come through, so my bad.
Reblogged this on MikeSight and commented:
Moments like that?
Reblogged this on NicoleKristeen17.
Reblogged this on totallypresent.
This is awesome, thanks for my thursday morning fun!
Thanks for reading!
Reblogged this on chunter's factor analysis.
Reblogged this on Many Worlds Theory and commented:
This is too perfect not share.
The part about markets reverting to the mean (as opposed to reacting to new info from the Fed) is economically illiterate and should be deleted or revised. Small moves in the market might legitimately be termed noise, though one agent’s noise is another’s signal. But mean reversion on a 1-day timescale would be an obvious, profitably exploitable regularity that would long since have been exploited away.
That’s an interesting point. Certainly it can’t be the case that the market always rebounds big immediately after a big fall.
That said, the point of the headline is more that shifts in the market may have no clear causal explanation, or at least not one grounded in a specific political event (like Bernanke saying something soothing). Thus, when the market rebounds, saying “Event X was the cause” may just be a nice post-hoc bedtime story we tell ourselves about a random event. Attributing the rebound to mean regression falls somewhat into the same trap, but at least honors the possibility that we’re dealing with a largely random process.
不错
A fraction of my friends will love this blog. 99/100ths of people I know will laugh at this.
Reblogged this on Imarunnerandsocanyou and commented:
I know this is a running blog, but I think all of you are mature enough to handle this material. Enjoy!