Estimating Economic Growth

The answer is 238.64%.  A good way of approximating is to use the rule of 70.  If x is the growth rate then the doubling time is approximately 70/x.  Thus, with a growth rate of 5% we expect a doubling (100% increase) in 14 years and a quadrupling in 28 years so a bit more than a tripling in 25 years (200% increase) is a good guess.

Comments

3.3% per doesn't seem like modest growth to me. That's ~38% per decade, I'd truly call that fast progress!

Rule of 70? I thought it was the rule of 72? I think 72 is more often closer to the correct factor than 70 is, and it also has more divisors, in particular 3, 4, 6, 8, 9 and 12 compared to 5 and 10 for 70.

Of course, in this particular instance, 5 was the one wanted... :)

A nice lecture about exponential growth is the video from Albert Bartlett:

http://www.globalpublicmedia.com/dr_albert_bartlett_arithmetic_population_and_energy

If you google it: it's easy to find all over the internet.

I don't think there has been a 1 to 1 relationship between wages and economic growth for quite some time. So perhaps people can be forgiven for underestimating a growth in other peoples prosperity when there own economic welfare grows at a comparatively sluggish pace. I think maybe the underestimating might be do to an empirical dampening. The right answer doesn't seem correct because it runs counter to peoples actual experiences.

For more on our systematic misunderstanding of exponential processes see "The Logic Of Failure: Recognizing And Avoiding Error In Complex Situations" by Dietrich Dorner

Isn't it just easier to look at the expontional growth of compound interest, i.e., 1.05^25?

Not to be a quibbler but we aren't even talking about exponential processes (right?)... compound growth is geometric. If we had exponential growth people would be even more astounded.

I've never driven 150 mph. But if you ask me how many miles I would travel if I drove 150 mph for 3 hours, I can answer that question.

You don't need personal experience in order to do math. I rather suspect that if people are underestimating the amount of growth 5% over 25 years brings, it is because they are bad at math, not because 2% growth seems more realistic to them.

With Google around you don't have to be good at math. Just type 1.05**25 into the tool bar.

Can I be incredulous that you are assuming an average population growth of 0% between now and 2100?

I'm sort of incredulous that people can suggest that a world average income of 200K (in constant dollars) is even possible. It seems to me that the real point of the exercise is to show that exponential growth is an implausible model for long term extrapolation.

mobile: Usually GDP growth is expressed as per capita income, which divides out the population. Also, as countries grow richer, population growth rates tend to decline rapidly, so if the world really does have $200,000 per capita income in 2100, it would not be unreasonable to expect a population growth rate of <1%.

FYI, the authors tried a 3% rate and it didn't make any difference (literally; this shows how bad people are at estimating). In fact, "The results show that for actual growth rates higher than 1%, both groups clearly underestimated growth..."

I'm sort of incredulous that people can suggest that a world average income of 200K (in constant dollars) is even possible. It seems to me that the real point of the exercise is to show that exponential growth is an implausible model for long term extrapolation.

Why are you incredulous? Think about what sort of goods were possessed by the richest in the world in the Middle Ages, or even 100 years ago-- the ability to have an orchestra play whatever music you wanted when you wanted, the ability to wear different clean clothes every day, the ability to travel, the ability to eat meat every day, etc. All of those things are within the grasp of even the poor in this country, even if things are not perfect. It is entirely reasonable to expect that everyone in the world would have access to wealth equal to that of someone making $200,000/year today, even if there will be still richer people.

I agree that these people are bad at math. But you and I know that global GDP won't be growing at a 5% real, per capita rate, so what was the point of asking the question, other than to snag them on their bad math?

One real point is that people since underestimate the power of compounding interest, they greatly disregard the value of, e.g., choosing policies that slow the rate of growth by 0.5% or 1.0% per annum. It doesn't sound like much, and on one year it's easily lost in measurement error. But over the years unsound policy really adds up.

So it's 1.05^25 - 1 times 100 to get it into a percentage. Why do we subtract the 1? i.e. why isn't it 338%?

The most powerful force in the universe is compound interest.
You subtract the 1 because "x% growth over t years" really means "principle times (1 + x/100) to the t-th power". For example, when you say "my income grew five percent last year", you really mean "my current income is equal to last year's income multiplied by 1.05".

Isn't it just easier to look at the expontional growth of compound interest, i.e., 1.05^25?
Which is easier to do in your head?

Not to be a quibbler but we aren't even talking about exponential processes (right?)... compound growth is geometric. If we had exponential growth people would be even more astounded.
Same thing. Geometric growth is the same as exponential growth; the former term is used for discrete cases (e.g., 1,2,4,8,16,...) and the latter for continuous situations (e.g., f(x) = 2^x).

and if you had $1 at the birth of Christ and it compounded at 1%/year, how much would you have today? What percentage of world wealth would you have at 3%/year?

What does this say about long term extrapolation?
Math exercise: 2009 / (100ln(2)/1.01) = 2009 * 1.01 / ln(2) = $500 million (doubles about 29 times, every 70 years).

Neal, thanks buddy. Good explanation. Another good explanation is that no growth is 1.0^25 -1, so of course you need to subtract 1.

@ Buzzcut

The '1' is subtracted out because we want the increase as opposed to the new total value. If you start with $1 and it grows at 5% for 25 years you get: $1*(1.05)^25 which as you correctly calculated equals $3.39 the new total value. In terms of growth we get: 100%*($3.39 - $1)/$1 = 239% -- short cutting the math gives the simplified growth formula of (((1+r)^t)-1)*100%

I've always heard rule of 72.
It's based on an interest rate of 8% annually compounded.

Note that 2=(1+x)^n => n=ln(2)/ln(1+x)
Plug in .08 for x and is 9.0=72/8

It sounds like this is just an issue with how the question is posed.. (I can only guess since I'm not going to subscribe to sciencedirect)

They appear to be asking a basic question about what will the number be in 25 years if the number grows 5% a year..

But, I would presume that people are assuming the question is about median wages/income.

eg, what will the median wage be for a 30 year old in 25 years if the GDP grows at 5% a year?

The correct answer is.. "who the fuck knows.." but you could also accept "I bet you $1000 that their income will not grow at 5% a year."

Insert lots of arguments about what income is.. or wages.. or what about intangible benefits like bigger televisions, more expensive healthcare coverage or vast numbers of attractive sexual partners.

Also, this is a neat article discussing gdp per capita and "average income":
read me!

Brad, off the top of my head, I'd hazard an uninformed guess that people cluster bigger numbers because they don't adequately plan out the chart. At least, that's the reason I do it. ;)

"I'm sort of incredulous that people can suggest that a world average income of 200K (in constant dollars) is even possible. It seems to me that the real point of the exercise is to show that exponential growth is an implausible model for long term extrapolation."

Why? What do people spend their first 200k/year on?

Shoes? Clothes? Food? Cars? Small boats? Homes? The cost of producing those things will continue to trend towards 0 even without cheap labor as more and more is done by robots and the robots get faster, better, and more numerous, and are able to do more and more things. http://www.youtube.com/watch?v=BKnXWMPbCyc

Healthcare? The costs of producing diagnostic machines will go down, drugs will become more effective, currently patented drugs will be generic, and there will be a much larger number of trained doctors. I find it completely plausible that by 2100 the average world citizen will be able to afford healthcare equivalent or better than the healthcare currently afforded by a person making $200,000 per year. The average US citizen certainly receives better healthcare than even the millionaires of 100 years ago.

Energy? This is the only real wildcard, but I would expect that in 2100 we will have solar power and/or fusion, and/or geothermal enough to supply massive amounts of clean energy.

fusion, 1% annual growth on $1 for 2000 years would yield roughly $440 million dollars. 3% annual growth would yield 4.7 x 10^25 dollars, which is more money than what currently exists on the planet.

People probably underestimated because the question was misleading. Median income per person has tended to grow at a much slower rate than overall GDP. If the country's economy grows by 5% per year, most of that money will be concentrated in the hands of the wealthy, so most people don't see any of that growth. They probably thought, well, every time GDP goes up, my income seems to stay the same, so this must be a trick question. Okay, they probably didn't think of that, and were just bad at math, but anything's possible.

John Pertz:

I divided at 1933, because ideologically that makes the beginning of "socialism", and at 1981 because that is the "restoration of free markets", Clinton's term because that's the time of crushing tax hikes that would doom the economy, and 2001 as the point where tax cuts would restore economic growth after the crushing Clinton tax burden.

Of course, 1860-1920 was an era of government policy supporting land redistribution that promoted economic development and industrialization, for example, the land give aways to the rail road companies, the opening of the west to farming to support industrialization in the east, and the rich mining opportunity nearly free government stolen land made so profitable.

Ok, I'm being snarky, but I want growth models tied to all the factors of production, land, resources, labor, technology, that are then backed up with good data.

As for "We have data sets on growth from the larger welfare states in France, Germany, and Italy. ... Unsurprisngly, they have permanently high rates of unemployment and sluggish rates of growth."

Well, you are rather vague in your statements, but I have compared unemployment data for EU nations to the US for "recent" years, and I see nothing to suggest their economies are performing much different. Yep, I know of the probloms of immigrant heritage communities and their endemic unemployment, but the social welfare system provides for them better than the equivalent communities in the US that suffer the same problems, specifically the blacks, the poor, and minority groups isolated in large communities in many areas with lack of transportation from where they live to the places where jobs exist.

And the younger people in the US are being shut out of the good jobs by the wonderful economy the 2001, 2003, 2006, 2008 tax cuts produced which generated an average of 1000 jobs a month during the whole of the Bush term, plus the older workers desperately hanging onto their jobs instead of retiring because they can't afford to retire. I think the problem in France has been described as one of the older workers staying in jobs protected by the social contract and thus preventing the young adults from finding opportunity in the workforce.

In any case, if the Democrats are like those European socialists, then the economic performance in Europe is probably better than if Republicans were running things, based on a comparison between decades of Republican vs Democratic leadership. Unless the argument you make is Europe is too much socialism, Republicans too littlr socialism, and the Democrats is just right.

Again being snarky. Theories should have some basis in reality because they should model reality, but their ability to model reality needs to be tested with supporting data.

Compound interest did wonders for the stock market last year.

Mulp

You are analyzing the data with ideological blinders on. You look at eras where there are democratic presidents and say "well there is growth there, so these guys must be correct."

I dont know why you do that. You need to bring things down to a more micro level. For instance, you can say that governments provide secondary public education and this is hugely beneficial for economic growth because XYZ public schools create competent workers and entrepreneurs for the economy.

Or you can say that governments have zoning laws, massive road building projects and these things have XYZ benefit for the economy.

Or you can say that governments spend tons of money funding the industrial military complex and this has XYZ benefit for the economy.

Or you can say that governments pay for the health care costs of the poor and elderly through medicaid/ medicare and this has XYZ benefit for the economy.

Or you can say that we are bailing out GM and the financial institutions and this is GREAT for economic growth for such and such reason.

However, to not tease out what private/public actions are responsible for the growth numbers, and simply declare that XYZ president was in office at the time of growth therefore he is responsible for growth, seems ideological.

As for your take on Italy, France, and Germany's unemployment issues, I am shocked. How did you look at the historical data over the past 20 years and say "Yep, even though France, Germany, and Italy have shown a clear tendency to have levels of unemployment at almost twice the U.S rate, I feel that there isnt much of a difference." LOL

Why on earth would you expect the average person to be able to compute compound interest over 25 periods in his head?? I am quite frankly very good at math, but I have never been called upon to do this type of calculation in my head, and so I was unaware of the handy approximation (rule of 70) for getting the doubling time. I can imagine that macroeconomists, mortgage bankers, and fixed-income asset experts might know this trick, but surely the rest of us are justifiably happy to be able to figure tip in our heads. I know there are other similar handy approximations, for things like square root and likely for logarithms, too. But, to use this case to say people are bad at math is to imply that being good at math is the same as being an effective replacement for a calculator. When I say I am good at math, I mean I can disect a numerical question into what calculation must be performed, and I can usually perform said calculation if it has a closed form (even calculus). I can usually approximate the answer to arbitrary accuracy if it doesn't involve any "special" functions. I will, nonetheless, have to rely on a calculator or pen and paper (the back of an envelope will do nicely) to get the numerical answer. This whole post strikes me as "people who don't know the same math tricks as me are dumb," which is really juvenile. It's telling that folks were arguing over whether it was the rule of 72 or 75, since "all" one has to do to figure out the "rule" is take ln(2) in your head! harumpf.

For investors here is what you need to know. You dont' hae to hit a home run, just don't lose money (on a multi-asset portfolio that is). assumptions made in most financial models is for positive growth in perpetuity, or some set number of years.

over 25yrs try adding a -5% in two years and a -2.5% in another, all equally spaced in time (9yrs at 5%, 10th yr at -5%, etc..) and do the math. It significantly reduces the growth. remember, it takes a 100% gain to make up for a 50% loss.

Also, you subtract the 1 b/c you added it to the .05 to begin with.

Russell, Russell, Russell ... *shakes head* Why not take the easy way?

Python 2.5.2 (r252:60911, Jul 31 2008, 17:28:52)
[GCC 4.2.3 (Ubuntu 4.2.3-2ubuntu7)] on linux2
Type "help", "copyright", "credits" or "license" for more information.
>>> (1.05**25 - 1)*100
238.63549408993885
>>>

"But, to use this case to say people are bad at math is to imply that being good at math is the same as being an effective replacement for a calculator"

The important part of the original post isn't that people were bad at estimating; it's that they SYSTEMATICALLY underestimated! In other words, the average person's intuition makes them underestimate the power of growth.

As far as the math wankery: yes, I'm a theoretical physicist so I don't expect others to be as good as I am at estimating. For what it's worth, I got 250% as the answer in my head in under 5 seconds. A few years ago I probably could have done better, but I've come to rely too much on computers for numerical estimates as a side effect of using them constantly for precision numerical work.

:)

And an average home will run $600k, but you will have spend three times that to live in an urban area and six times that for nice one.

Wolfram Alpha, for so long the butt of certain individuals' humour, finally comes up trumps! Entering (after a couple of false starts) "value after 25 years with 5% growth" it gave me a long table with the answer in the bottom left cell (corrected for the fact that it includes the original 100%, so that it says 3.386 instead of 2.386). Hurrah! It also copes concisely with 1.05**25.

Hmm, if anything, this shows how right that nice Mr Obama is to want to borrow from the future, where everybody is rich :))) <ducks...>

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