Missing Magnitudes

Cell Phone Charger Leakage

I had a friend once point out that cell phone chargers use electricity when they're unplugged. This is true, of course, and intuitive to anyone with a basic understanding of electronics. A simple model of electrical circuits predicts that an electrical current will follow the path of least resistance. But what happens if there is no 'path'?

Think of this another way: there is always a path from the positive to the negative pole. It's just that sometimes that path is blocked by something difficult to pass through. Even for a normal outlet, there's air between the positive and negative poles. The air is extremely difficult to pass through, but not impossible, so a little power will find its way even through the open air. (This happens at large scales with lightning, where millions of volts and long distances are involved.) But in the case of a cell phone charger, those two poles are brought close together (to keep the plug small and the phone small). That means there's less air, and therefore less resistance. We might call this 'leaky current', in the sense that some tiny amount of electricity 'leaks' through your cell phone charger even when it's not plugged into the phone, so long as it is plugged into the wall outlet.

The amount of energy that leaks from your cell phone charger is about 0.1 to 0.5 watts per hour. My friend compared that to a leaky faucet, and using that analogy suggested I unplug my phone when it's not in use. As I would fix a leaky faucet, I should fix a leaky charger. He insisted that, although the amount of electricity I saved might sound small, the real savings would accumulate if everyone followed this same advice. Just think about all the electricity people were wasting, and how much we could save from this simple procedure!

Of course I didn't do that, and the reason is the subject of today's post: Missing Magnitudes. There's a failure mode in logic, where people will be right about a thing in theory, but where the practical application forgets to calculate the magnitude appropriately. Once you calculate the magnitude, it becomes obvious that the person is 'right' about the thing, but in a way that doesn't matter - or in a way that doesn't match their interest in the matter.

Let's start with the cell phone charger example. We'll take the high end of 0.5 watts per hour, assuming I have a particularly leaky (cheap) cell phone charger. 0.5 watts * 24 hours/day * 365.25 days/year = 4,383 watt-hours. That's not quite 4.4 kWh. (I got an extra 3 watts by calculating in each year's portion of a leap day!)

How does that compare at a large scale? Last month, my solar system generated about 1.3 megawatt-hours. In other words, every day, my solar panels generated ten times more electricity that the annual savings of unplugging a cell phone charger. Whether or not we have an extra passing cloud could be the same difference as the cell phone charger's usage.

But wait! The claim is that this will add up. Once we aggregate the savings over the whole population doesn't this become a major power saving idea?


Doing the math is trivial here (just add up the number of people in your city, country, the world - then divide by the total electricity usage of all those people), but the intuition is quicker and easier so that's what I want to focus on. In this scenario, it's the percentages that matter. If this energy savings is a tiny fraction of a percentage of my personal energy usage, then when we aggregate the savings we're going to be aggregating it as a percentage of everyone's usage and it will still be a tiny fraction of a percent. A public campaign to encourage cell phone charger unplugging habits would literally cost more energy than forgetting the whole idea.

Petty Cash

This reminds me of a company policy I once ran into, where the company didn't require a receipt or expense report for any expense below a specific limit (I think it was $50). The rationale was that they'd run the calculations and discovered that the majority of their expense reports were for these small-item purchases, and that there was a cost to process the expense reports. They figured the amount of money they saved in eliminating expense reports for small-ticket items was greater than the amount of money they expected to lose from employees who abused the system, buying unjustified items under the dollar limit (even a few personal items).

I'm not sure how well that kind of policy would transfer if applied to every company. Certainly, if you have a highly untrustworthy workforce, it will be difficult to make back your expense report money when employees go out to lunch every day on the company dime. The point is that for this company, they discovered that the concern - unjustified expenses - was lower than the cost to fix the concern. They didn't stop by asking, "Are employees going to make unjustified expenses unless we track the receipts?" They asked, "How big is the problem? How much would it take to fix that problem? Is the remedy worth the cost?"

I see this issue in many debates, where one side will be right in every way ... except that they never considered the magnitude. In some cases, the magnitude is unknown - or even guessed at - yet the participants in the debate focus on who is right, rather than on the substantive discussion about magnitudes.


An example where we see this play out in public debate is around masks and social distancing. I was disappointed recently when I read Scott Alexander's (does he still go by the pseudonym?) recent write-up on whether 'lockdowns' work. The post was long and, ultimately, meaningless, because he failed to address the most important question: what's the magnitude compared to the seasonal effect? As I've pointed out in this space a few times, the seasonal effect of COVID-19 is large. If we learned anything back in November 2020, it was that none of the social distancing, mask, lockdown, or other policies mattered in relationship to the seasonal effect. Across the US, and without regard to policy or population size, COVID-19 outbreaks surged in late October/early November. The surge in cases persisted until spring, when they went away. As I already shown, the spring drawdown didn't even have anything to do with the vaccine, but had everything to do with the season!

Yet Scott spent many hours, thousands of words, lots of graphs, and a lot of mental effort agonizing over what human efforts are effective, and to what extent. He might have saved himself a lot of grief by simply asking the question, 'Is the magnitude meaningful?'

Minimum Wage

Another example might be found in economics (warning: completely outside my field of expertise). From what I've read, there are many studies on one side of the debate over minimum wage showing increased unemployment when the minimum wage is raised. But there are also a lot of other studies showing no negative effect when the minimum wage is raised. People line up on either side of this issue, and the fight can get pretty intense.

But what's the magnitude of the problem here? The percent of workers making minimum wage is around 2.3%, meaning few workers are taking these jobs - even at the entry level, or to build up skills, or [insert free-market libertarian argument here] - because they have better options. Maybe that's because the federal minimum wage hasn't moved anywhere in years: $7.25/hr.

The federal minimum wage was last raised in 2009. Since then, inflation has effectively reduced the minimum wage, such that to keep up with what it was in 2009 it would have to be raised by about $2/hr. today. What have workers done since the last time it was raised? There's a concept in economics called a 'reserve wage', where someone is likely to just stay unemployed - or keep looking for work - if they are only offered a job lower than a certain dollar amount. If you're a lawyer who has passed the bar, you're not going to take a job at the grocery store just because you're between legal firms. Even if you only plan to work at the grocery store between now and six months from now when your new law firm is ready for you to join, it doesn't seem worth it to work at the grocery store. Partly it's professional (you feel like you left the entry-level market behind when you went into law school), but partly it's because the magnitudes are so different between what you can make as a cashier versus what you can make at the law firm - even though you're not working there at the moment. If you're billing at $200/hr. two hours of legal work are the equivalent of working a whole week at the grocery store for $10/hr.

But this reserve wage doesn't just apply to people at the top of the income ladder. It also applies at the bottom. Who would be willing to work for twenty cents per day? Maybe interns, but then they're not really working for the money so much as the experience. For interns the wage isn't even a factor. For everyone else, it is. So the question we should be asking is, how many people have a reservation wage - a floor below which they won't go - that's below the minimum wage? Everyone with a reservation wage above the minimum we would expect to go unemployed rather than take the minimum wage job, thus a minimum wage is moot for that person. It might be set at twenty cents per day or twenty cents per hour and not make any difference.

This might explain why some researchers see a minimum wage effect, while others don't. When local politicians raise prices above the reservation wage of a significant number of people those people are affected, because a minimum wage below the reservation wage can't impact that person's employment. If you're doing a study where the minimum wage is raised from $7.25/hr. to $9.25/hr., you're not doing anything more than moving back to 2009 levels. If your study looks at a rise from $7.25/hr. to $75/hr. you're going to see a strong impact as a result of the policy.

Why is this question important? Let's say you're opposed to the minimum wage because you're convinced by the basic economics argument that when you raise the price of something you decrease demand for that thing. But if minimum wage is $2 less than it was 12 years ago, you might assume that the magnitude of the effect of changing the policy to adjust up by $2/hr. will be negligible - as in those studies that didn't find an effect from minimum wage increases. There are a lot of political bargains you could make with politicians whose political careers are dependent on raising the minimum wage. You could easily give up a meaningless $2/hr. in exchange for something meaningful to you. Filling in the missing magnitudes guides you to that insight.

Climate Change

Remembering to check the magnitude can also help orient a debate away from unproductive right versus wrong. Sometimes you're right, but it doesn't matter. Other times you're right, and it matters, but you need to do more than convince the other person of the accuracy of the principles you're working from. Rather, you need to convince them that the magnitude matters. This can be difficult, because the other person may not even realize this is their underlying concern, such that you go round and round in circles debating whether when you should really be discussing how much.

In no other policy debate do I see this problem more profoundly than in the public discourse surrounding climate change. (Warning: also well outside my field. When people ask me what my opinion is about climate change I tell them I don't know about it, I have no training in the discipline, they should discuss it with someone who does. I do know about philosophy of science, though.)

Allow me to indulge in a little categorization of the different groups of people I observe in public debates about this topic (warning - overgeneralizations):

Deniers: This group of people are uninterested in the theory about climate change or whether it's even possible for humans to have an impact on the planet. They're focused on whether the phenomenon is happening, more than on the magnitudes in question.

Skeptics: Yes, this group is different than the deniers. If you listen closely, they do not object to the theoretical idea that the climate might warm due to human activity, but they're skeptical that the magnitude is large enough to have a measurable - or meaningful - impact on the climate. Some say there can be no meaningful impact no matter how much CO2 is emitted. Others say it's not enough to worry about anytime in the near future, and all policy surrounding climate change is chasing something that's too minimal to matter.

Slow warmers: This group also agrees to the theoretical idea. They believe in a magnitude for human-caused warming that is gradual, but is also concerning over long-term trends. They're willing to go along with climate mitigation strategies so long as they aren't too strict.

Medium warmers: This group thinks the magnitudes are small, but not as small as the slow warmers. They see the cumulative effects over decades as a significant concern that will change the face of the planet is undesirable ways if nothing is done to arrest the warming sooner rather than later.

Catastrophic climate changers: This group believes that the effects of climate change are of a magnitude at the far end of the scale. They stress immediate action, because they perceive the negative outcomes to be imminent and catastrophic. Some use the phrase 'existential threat'.

Most of the back-and-forth I see on the internet surrounding climate change is about whether it's happening. There's a lot of anger generated surrounding this question. But, outside a few unpersuadable outright deniers, the question of magnitude seems to be where most disagreement on this subject lies. Yet how much of the discourse surrounds the magnitude question? I see very little. Again, I'm not interested here in endorsing a particular side. Cheerleading from outside a scientific discipline that I have no part of isn't that interesting to me. What is interesting is how little progress is being made because of a failure to focus on magnitudes.

I see the problem of Missing Magnitudes a lot. My hope is that you'll start seeing it more, too, and that seeing it will be enough to help you add that magnitude back in - or at least ask for it. The next time someone approaches you with an outlandish claim, don't just ask yourself whether they're right; ask whether it even matters.


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